Denote the complete elliptic integral of the first kind by $$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$ and $$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$
Question: Given a real $x$, $0\lt x\lt 1$, is there an algorithm which determines whether $f(x)$ is positive rational?
Thoughts: Let $\lambda^*(x)=\sqrt{\lambda (i\sqrt{x})}$ where $\lambda$ is the modular lambda function: $$\lambda (x)=\left(\frac{\sum_{n=-\infty}^\infty e^{\pi i x\left(n+\frac{1}{2}\right)^2}}{\sum_{n=-\infty}^\infty e^{\pi ixn^2}}\right)^4$$ It turns out that $\lambda^*$ is the inverse of $f$: $$\lambda^*(f(x))=x,\quad f(\lambda^*(x))=x$$
It is known that $\lambda^*(f(x))$ is algebraic if $f(x)$ is positive rational. Equivalently, $x$ is algebraic if $f(x)$ is positive rational. By contrapositive,
If $x$ is not algebraic, then $f(x)$ is not positive rational.
So, a part of the problem is already solved. It remains to solve
If $x$ is algebraic, can we determine whether $f(x)$ is positive rational?
The answer is known for some algebraic $x$, for example $$f\left((\sqrt{10}-3)(\sqrt{2}+1)^2\right)=\frac{2}{5}.$$
This question was also asked on MSE (https://math.stackexchange.com/questions/4247055/determine-whether-frack-left-sqrt1-x2-right2kx2-is-positive-rati).
